Question

A drawer contains $5$ brown and $4$ blue socks well mixed. A man reaches the drawer and pulls out $2$ socks at random. What is the probability that they match ?

• A. $\frac{2}{9}$
• B. $\frac{1}{3}$
• C. $\frac{4}{9}$
• D. $\frac{5}{9}$

Answer 1

The correct option is C

$\frac{4}{9}$

Explanation for the correct option:

Finding the required probability:

Total no. of socks $=5+4=9$

No. of socks pulled out $=2$

Let $S$ be the no. of ways in which two socks can be pulled out.

$\begin{array}{rcl}& \Rightarrow & n\left(S\right)={}^{9}C_2\\ & =& \frac{9!}{2!7!}\left[{\because }^n{C}_r=\frac{n!}{r!\left(n-r\right)!}\right]\\ & =& \frac{9\times 8\times 7!}{2\times 7!}\\ & =& 36\end{array}$

We have to find the probability that the socks should match, which means either both socks are brown or both are blue.

Let $A$ be the event that both socks are either brown or blue,

$\begin{array}{rcl}& \Rightarrow & n\left(A\right)={}^{5}C_2+{}^{4}C_2\\ & =& \frac{5!}{2!3!}+\frac{4!}{2!2!}\\ & =& \left(\frac{5\times 4\times 3!}{2\times 3!}\right)+\frac{4\times 3\times 2!}{2\times 2!}\\ & =& 10+6\\ & =& 16\end{array}$

Therefore, the required probability:

$\begin{array}{rcl}P\left(A\right)& =& \frac{n\left(A\right)}{n\left(S\right)}\\ & =& \frac{16}{36}\\ & =& \frac{4}{9}\end{array}$

Hence, the correct answer is option C.